Abstract

Analytical expressions of the nonparaxial propagation of even and odd elegant Laguerre-Gaussian beams orthogonal to the optical axis of a uniaxial crystal are derived. The intensity distributions of even and odd elegant Laguerre-Gaussian beams and their three components propagating in uniaxial crystals orthogonal to the optical axis are illustrated by numerical examples. Even though one of the two transversal components of even and odd elegant Laguerre-Gaussian beams in the input plane is set to be zero, this transversal and the longitudinal components have nonzero intensities and cannot be neglected upon propagation inside the uniaxial crystal. The evolution laws of even and odd elegant Laguerre-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis are also demonstrated. The intensity distributions of even and odd elegant Laguerre-Gaussian beams can be modulated by the uniaxial crystal, which is beneficial to the some applications involving in the special beam profile.

© 2012 OSA

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2012 (1)

2011 (3)

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 53(2), 20701 (2011).
[CrossRef]

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre-Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011).
[CrossRef]

H. Xu, Z. Cui, and J. Qu, “Propagation of elegant Laguerre-Gaussian beam in non-Kolmogorov turbulence,” Opt. Express 19(22), 21163–21173 (2011).
[CrossRef] [PubMed]

2010 (5)

W. Nasalski, “Elegant Hermite-Gaussian and Laguerre-Gaussian beams at a dielectric interface,” Opt. Appl. 40, 615–622 (2010).

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre-Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
[CrossRef]

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” PIER 103, 33–56 (2010).
[CrossRef]

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57(5), 375–384 (2010).
[CrossRef]

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

2009 (5)

Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express 17(17), 14865–14871 (2009).
[CrossRef] [PubMed]

D. Liu and Z. Zhou, “Generalized stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 11(6), 065710 (2009).
[CrossRef]

B. Tang, “Hermite-cosine-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26(12), 2480–2487 (2009).
[CrossRef] [PubMed]

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 54(1), 95–101 (2009).
[CrossRef]

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
[CrossRef] [PubMed]

2008 (2)

A. April, “Nonparaxial elegant Laguerre-Gaussian beams,” Opt. Lett. 33(12), 1392–1394 (2008).
[CrossRef] [PubMed]

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

2007 (3)

2005 (1)

Y. Pagani and W. Nasalski, “Diagonal relations between elegant Hermite-Gaussian and Laguerre-Gaussian beam fields,” Opto-Electron. Rev. 13, 51–60 (2005).

2004 (5)

M. A. Bandres and J. C. Gutiérrez-Vega, “Higher-order complex source for elegant Laguerre-Gaussian waves,” Opt. Lett. 29(19), 2213–2215 (2004).
[CrossRef] [PubMed]

Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21(12), 2375–2381 (2004).
[CrossRef] [PubMed]

D. Zhao, Z. Mei, J. Gu, H. Mao, L. Chen, and S. Wang, “Propagation characteristics of truncated standard and elegant Laguerre-Gaussian beams,” Proc. SPIE 5639, 149–158 (2004).
[CrossRef]

Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004).
[CrossRef]

B. Lü and S. Luo, “Propagation properties of three-dimensional flatted Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36(1), 51–56 (2004).
[CrossRef]

2003 (1)

2002 (2)

A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19(4), 792–796 (2002).
[CrossRef] [PubMed]

S. Luo and B. Lü, “Propagation of the kurtosis parameter of elegant Hermite-Gaussian and Laguerre-Gaussian beams passing through ABCD systems,” Optik (Stuttg.) 113(5), 227–231 (2002).
[CrossRef]

2001 (3)

1998 (1)

S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45(10), 1999–2009 (1998).
[CrossRef]

1993 (1)

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[CrossRef]

1986 (1)

E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. 3(4), 465–469 (1986).
[CrossRef]

1985 (1)

April, A.

Bandres, M. A.

Baykal, Y. K.

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” PIER 103, 33–56 (2010).
[CrossRef]

Borghi, R.

Cai, Y.

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” PIER 103, 33–56 (2010).
[CrossRef]

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre-Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
[CrossRef]

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57(5), 375–384 (2010).
[CrossRef]

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
[CrossRef] [PubMed]

Chen, F.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 53(2), 20701 (2011).
[CrossRef]

Chen, L.

D. Zhao, Z. Mei, J. Gu, H. Mao, L. Chen, and S. Wang, “Propagation characteristics of truncated standard and elegant Laguerre-Gaussian beams,” Proc. SPIE 5639, 149–158 (2004).
[CrossRef]

Chen, R.

Chen, Y.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 53(2), 20701 (2011).
[CrossRef]

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

Chu, X.

Ciattoni, A.

Cincotti, G.

Cui, Z.

H. Xu, Z. Cui, and J. Qu, “Propagation of elegant Laguerre-Gaussian beam in non-Kolmogorov turbulence,” Opt. Express 19(22), 21163–21173 (2011).
[CrossRef] [PubMed]

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre-Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011).
[CrossRef]

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre-Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
[CrossRef]

Deng, D.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

D. Deng, Q. Guo, L. Wu, and X. Yang, “Propagation of radially polarized elegant light beams,” J. Opt. Soc. Am. B 24(3), 636–643 (2007).
[CrossRef]

Elias, L. R.

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[CrossRef]

Eyyuboglu, H. T.

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” PIER 103, 33–56 (2010).
[CrossRef]

Fan, Z.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

Fukumitsu, O.

Gu, J.

Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express 17(17), 14865–14871 (2009).
[CrossRef] [PubMed]

D. Zhao, Z. Mei, J. Gu, H. Mao, L. Chen, and S. Wang, “Propagation characteristics of truncated standard and elegant Laguerre-Gaussian beams,” Proc. SPIE 5639, 149–158 (2004).
[CrossRef]

Guo, Q.

Gutiérrez-Vega, J. C.

Kimel, I.

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[CrossRef]

Korotkova, O.

Li, J.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 53(2), 20701 (2011).
[CrossRef]

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

Liu, D.

D. Liu and Z. Zhou, “Generalized stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 11(6), 065710 (2009).
[CrossRef]

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 54(1), 95–101 (2009).
[CrossRef]

Lü, B.

B. Lü and S. Luo, “Propagation properties of three-dimensional flatted Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36(1), 51–56 (2004).
[CrossRef]

S. Luo and B. Lü, “Propagation of the kurtosis parameter of elegant Hermite-Gaussian and Laguerre-Gaussian beams passing through ABCD systems,” Optik (Stuttg.) 113(5), 227–231 (2002).
[CrossRef]

Luo, S.

B. Lü and S. Luo, “Propagation properties of three-dimensional flatted Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36(1), 51–56 (2004).
[CrossRef]

S. Luo and B. Lü, “Propagation of the kurtosis parameter of elegant Hermite-Gaussian and Laguerre-Gaussian beams passing through ABCD systems,” Optik (Stuttg.) 113(5), 227–231 (2002).
[CrossRef]

Mao, H.

D. Zhao, Z. Mei, J. Gu, H. Mao, L. Chen, and S. Wang, “Propagation characteristics of truncated standard and elegant Laguerre-Gaussian beams,” Proc. SPIE 5639, 149–158 (2004).
[CrossRef]

Mei, Z.

Z. Mei and J. Gu, “Comparative studies of paraxial and nonparaxial vectorial elegant Laguerre-Gaussian beams,” Opt. Express 17(17), 14865–14871 (2009).
[CrossRef] [PubMed]

Z. Mei, “The elliptical elegant Laguerre-Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” Optik (Stuttg.) 118(8), 361–366 (2007).
[CrossRef]

Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004).
[CrossRef]

D. Zhao, Z. Mei, J. Gu, H. Mao, L. Chen, and S. Wang, “Propagation characteristics of truncated standard and elegant Laguerre-Gaussian beams,” Proc. SPIE 5639, 149–158 (2004).
[CrossRef]

Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21(12), 2375–2381 (2004).
[CrossRef] [PubMed]

Nasalski, W.

W. Nasalski, “Elegant Hermite-Gaussian and Laguerre-Gaussian beams at a dielectric interface,” Opt. Appl. 40, 615–622 (2010).

Y. Pagani and W. Nasalski, “Diagonal relations between elegant Hermite-Gaussian and Laguerre-Gaussian beam fields,” Opto-Electron. Rev. 13, 51–60 (2005).

Pagani, Y.

Y. Pagani and W. Nasalski, “Diagonal relations between elegant Hermite-Gaussian and Laguerre-Gaussian beam fields,” Opto-Electron. Rev. 13, 51–60 (2005).

Palma, C.

Piper, J. A.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
[CrossRef]

Porras, M. A.

Qu, J.

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre-Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011).
[CrossRef]

H. Xu, Z. Cui, and J. Qu, “Propagation of elegant Laguerre-Gaussian beam in non-Kolmogorov turbulence,” Opt. Express 19(22), 21163–21173 (2011).
[CrossRef] [PubMed]

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre-Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
[CrossRef]

Saghafi, S.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45(10), 1999–2009 (1998).
[CrossRef]

Santarsiero, M.

Shao, J.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

Sheppard, C. J. R.

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Opt. 45(10), 1999–2009 (1998).
[CrossRef]

Shi, J.

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre-Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011).
[CrossRef]

Takenaka, T.

Tang, B.

Wang, F.

F. Wang, Y. Cai, H. T. Eyyuboglu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” PIER 103, 33–56 (2010).
[CrossRef]

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
[CrossRef] [PubMed]

Wang, S.

D. Zhao, Z. Mei, J. Gu, H. Mao, L. Chen, and S. Wang, “Propagation characteristics of truncated standard and elegant Laguerre-Gaussian beams,” Proc. SPIE 5639, 149–158 (2004).
[CrossRef]

Wang, Y.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 53(2), 20701 (2011).
[CrossRef]

Wu, L.

Xin, Y.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 53(2), 20701 (2011).
[CrossRef]

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

Xu, H.

Xu, S.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 53(2), 20701 (2011).
[CrossRef]

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

Yang, X.

Yokota, M.

Yu, H.

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[CrossRef]

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C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57(5), 375–384 (2010).
[CrossRef]

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Z. Mei and D. Zhao, “Propagation of Laguerre-Gaussian and elegant Laguerre-Gaussian beams in apertured fractional Hankel transform systems,” J. Opt. Soc. Am. A 21(12), 2375–2381 (2004).
[CrossRef] [PubMed]

D. Zhao, Z. Mei, J. Gu, H. Mao, L. Chen, and S. Wang, “Propagation characteristics of truncated standard and elegant Laguerre-Gaussian beams,” Proc. SPIE 5639, 149–158 (2004).
[CrossRef]

Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004).
[CrossRef]

Zhao, Q.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 53(2), 20701 (2011).
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Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre-Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011).
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[CrossRef]

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Zhou, M.

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 53(2), 20701 (2011).
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[CrossRef]

Appl. Phys. B (1)

Y. Zhong, Z. Cui, J. Shi, and J. Qu, “Polarization properties of partially coherent electromagnetic elegant Laguerre-Gaussian beams in turbulent atmosphere,” Appl. Phys. B 102(4), 937–944 (2011).
[CrossRef]

Eur. Phys. J. Appl. Phys. (1)

J. Li, Y. Xin, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, and F. Chen, “Diffraction of Gaussian vortex beam in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. Appl. Phys. 53(2), 20701 (2011).
[CrossRef]

Eur. Phys. J. D (2)

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 54(1), 95–101 (2009).
[CrossRef]

J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 57(3), 419–425 (2010).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Z. Mei and D. Zhao, “The generalized beam propagation factor of truncated standard and elegant Laguerre-Gaussian beams,” J. Opt. A, Pure Appl. Opt. 6(11), 1005–1011 (2004).
[CrossRef]

D. Liu and Z. Zhou, “Generalized stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 11(6), 065710 (2009).
[CrossRef]

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E. Zauderer, “Complex argument Hermite-Gaussian and Laguerre-Gaussian beams,” J. Opt. Soc. Am. 3(4), 465–469 (1986).
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J. Opt. Soc. Am. A (7)

J. Opt. Soc. Am. B (1)

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W. Nasalski, “Elegant Hermite-Gaussian and Laguerre-Gaussian beams at a dielectric interface,” Opt. Appl. 40, 615–622 (2010).

Opt. Commun. (3)

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre-Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
[CrossRef]

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281(2), 202–209 (2008).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, and J. A. Piper, “Characterising elegant and standard Hermite-Gaussian beam modes,” Opt. Commun. 191(3-6), 173–179 (2001).
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Proc. SPIE (1)

D. Zhao, Z. Mei, J. Gu, H. Mao, L. Chen, and S. Wang, “Propagation characteristics of truncated standard and elegant Laguerre-Gaussian beams,” Proc. SPIE 5639, 149–158 (2004).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Contour graph of the intensity of the x-component of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.

Fig. 2
Fig. 2

Contour graph of the intensity of the y-component of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.

Fig. 3
Fig. 3

Contour graph of the intensity of the longitudinal component of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.

Fig. 4
Fig. 4

Contour graph of the intensity of an even elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.

Fig. 5
Fig. 5

Contour graph of the intensity of an even elegant Laguerre-Gaussian beam propagating in the observation plane z = 3zr of different uniaxial crystals. (a) e = 0.6. (b) e = 0.8. (c) e = 1.0. (d) e = 1.2.

Fig. 6
Fig. 6

Contour graph of the intensity of the x-component of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.

Fig. 7
Fig. 7

Contour graph of the intensity of the y-component of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.

Fig. 8
Fig. 8

Contour graph of the intensity of the longitudinal component of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.

Fig. 9
Fig. 9

Contour graph of the intensity of an odd elegant Laguerre-Gaussian beam in several observation planes of the uniaxial crystal. e = 1.5. (a) z = 0.1zr. (b) z = zr. (c) z = 3zr. (d) z = 5zr.

Fig. 10
Fig. 10

Contour graph of the intensity of an odd elegant Laguerre-Gaussian beam propagating in the observation plane z = 3zr of different uniaxial crystals. (a) e = 0.6. (b) e = 0.8. (c) e = 1.0. (d) e = 1.2.

Equations (29)

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ε=( n e 2 0 0 0 n o 2 0 0 0 n o 2 ),
[ E x ( ρ 0 ,0) E y ( ρ 0 ,0) ]=[ ( ρ 0 w 0 ) m L n m ( ρ 0 2 w 0 2 )exp( ρ 0 2 w 0 2 )cos(m φ 0 ) 0 ],
ρ 0 m L n m ( ρ 0 2 )cos(m φ 0 )= (1) n 2 2n+m n! l=0 n s=0 [m/2] (1) s ( n l )( m 2s ) H M 1 ( x 0 ) H N 1 ( y 0 ),
[ E x ( ρ 0 ,0) E y ( ρ 0 ,0) ]=[ (1) n 2 2n+m n! l=0 n s=0 [m/2] (1) s ( n l )( m 2s ) H M 1 ( x 0 w 0 ) H N 1 ( y 0 w 0 )exp( ρ 0 2 w 0 2 ) 0 ].
E(ρ,z)= d 2 kexp(ikρ)exp(i k ez z)( E ˜ x (k) k x k y k 0 2 n o 2 k x 2 E ˜ x (k) k ez k x k 0 2 n o 2 k x 2 E ˜ x (k) ) + d 2 kexp(ikρ)exp(i k oz z)( 0 k x k y k 0 2 n o 2 k x 2 E ˜ x (k)+ E ˜ y (k) k y k oz [ k x k y k 0 2 n o 2 k x 2 E ˜ x (k)+ E ˜ y (k) ] ) ,
E ˜ j (k)= 1 (2π) 2 E j ( ρ 0 ,0)exp[ i( k x x 0 + k y y 0 ) ]d x 0 d y 0 ,
k ez = [ k o 2 n e 2 ( n e 2 / n o 2 ) k x 2 k y 2 ] 1/2 , k oz = ( k o 2 n 0 2 k x 2 k y 2 ) 1/2 .
E(ρ,z)=exp(i k 0 n e z) d 2 kexp(ikρ)exp( i n e 2 k x 2 + n o 2 k y 2 2 k 0 n e n o 2 z )( E ˜ x (k) k x k y k 0 2 n o 2 E ˜ x (k) n e k x k 0 n o 2 E ˜ x (k) ) +exp(i k 0 n o z) d 2 kexp(ikρ)exp( i k x 2 + k y 2 2 k 0 n o z )( 0 k x k y k 0 2 n o 2 E ˜ x (k)+ E ˜ y (k) k y k o n o E ˜ y (k) ) .
E x (ρ,z)= k 0 n o 2πiz E x ( ρ 0 ,0) Λ e (ρ, ρ 0 )d x 0 d y 0 ,
E y (ρ,z)= i k 0 n o 2π z 3 E x ( ρ 0 ,0)(x x 0 )(y y 0 )[ Λ e (ρ, ρ 0 ) Λ o (ρ, ρ 0 )]d x 0 d y 0 + k 0 n o 2πiz E y ( ρ 0 ,0) Λ o (ρ, ρ 0 )d x 0 d y 0 ,
E z (ρ,z)= i k 0 n o 2π z 2 [ E x ( ρ 0 ,0)(x x 0 ) Λ e (ρ, ρ 0 )+ E y ( ρ 0 ,0)(y y 0 ) Λ o (ρ, ρ 0 ) ]d x 0 d y 0 ,
Λ e (ρ, ρ 0 )=exp(i k 0 n e z)exp{ k 0 2iz n e [ n o 2 (x x 0 ) 2 + n e 2 (y y 0 ) 2 ] },
Λ o (ρ, ρ 0 )=exp(i k 0 n o z)exp{ k 0 n o 2iz [ (x x 0 ) 2 + (y y 0 ) 2 ] }.
H M (x) exp[ (xy) 2 /α]dx= πα (1α) M/2 H M [y (1α) 1/2 ],
x H M (x)= 1 2 H M+1 (x)+M H M1 (x),
E x (ρ,z)= (1) n k 0 n o 2 2n+m+1 n!iπz exp(ie k 0 n o z) l=0 n s=0 [m/2] (1) s ( n l )( m 2s ) T M 1 (x) U N 1 (y),
E y (ρ,z)= (1) n i k 0 n o 2 2n+m+1 n!π z 3 l=0 n s=0 [m/2] (1) s ( n l )( m 2s ){exp(ie k 0 n o z){xy T M 1 (x) U N 1 (y) w 0 x × T M 1 (x)[0.5 U N 1 +1 (y)+ N 1 U N 1 1 (y)] w 0 y U N 1 (y)[0.5 T M 1 +1 (x)+ M 1 T M 1 1 (x)] + w 0 2 [0.5 T M 1 +1 (x)+ M 1 T M 1 1 (x)][0.5 U N 1 +1 (y)+ N 1 U N 1 1 (y)]}exp(i k 0 n o z){xy × V M 1 (x) V N 1 (y) w 0 x V M 1 (x)[0.5 V N 1 +1 (y)+ N 1 V N 1 1 (y)] w 0 y V N 1 (y)[0.5 V M 1 +1 (x) + M 1 V M 1 1 (x)]+ w 0 2 [0.5 V M 1 +1 (x)+ M 1 V M 1 1 (x)][0.5 V N 1 +1 (y)+ N 1 V N 1 1 (y)]}},
E z (ρ,z)= (1) n i k 0 n o 2 2n+m+1 n!π z 2 exp(ie k 0 n o z) l=0 n s=0 [m/2] (1) s ( n l )( m 2s ){x T M 1 (x) U N 1 (y) w 0 U N 1 (y) × [0.5 T M 1 +1 (x)+ M 1 T M 1 1 (x)]},
T μ (x)= w 0 πα (1α) μ/2 exp(a x 2 ) H μ [ α z r n o x iez w 0 (1α) 1/2 ],
U μ (y)= w 0 πβ (1β) μ/2 exp(b y 2 ) H μ [ β z r e n o y iz w 0 (1β) 1/2 ],
V μ (j)= w 0 πγ (1γ) μ/2 exp(c j 2 ) H μ [ γ z r n 0 j iz w 0 (1γ) 1/2 ],
e= n e n o , z r = k 0 w 0 2 2 ,α= ( 1 i z r n o ez ) 1 ,β= ( 1 ie z r n o z ) 1 ,γ= ( 1 i z r n 0 z ) 1 ,
a=α ( z r n o iez w 0 ) 2 + k 0 n o 2iez ,b=β ( e z r n o iz w 0 ) 2 + e k 0 n o 2iz ,c=γ ( z r n o iz w 0 ) 2 + k 0 n o 2iz .
[ E x ( ρ 0 ,0) E y ( ρ 0 ,0) ]=[ ( ρ 0 w 0 ) m L n m ( ρ 0 2 w 0 2 )exp( ρ 0 2 w 0 2 )sin(m φ 0 ) 0 ].
ρ 0 m L n m ( ρ 0 2 )sin(m φ 0 )= (1) n 2 2n+m n! l=0 n s=0 [(m1)/2] (1) s ( n l )( m 2s+1 ) H M 1 1 ( x 0 ) H N 1 +1 ( y 0 ),
[ E x ( ρ 0 ,0) E y ( ρ 0 ,0) ]=[ (1) n 2 2n+m n! l=0 n s=0 [(m1)/2] (1) s ( n l )( m 2s+1 ) H M 1 1 ( x 0 w 0 ) H N 1 +1 ( y 0 w 0 )exp( ρ 0 2 w 0 2 ) 0 ].
E x (ρ,z)= (1) n k 0 n o 2 2n+m+1 n!iπz exp(ie k 0 n o z) l=0 n s=0 [(m1)/2] (1) s ( n l )( m 2s+1 ) T M 1 1 (x) U N 1 +1 (y),
E y (ρ,z)= (1) n i k 0 n o 2 2n+m+1 n!π z 3 l=0 n s=0 [(m1)/2] (1) s ( n l )( m 2s+1 ){exp(ie k 0 n o z){xy T M 1 1 (x) U N 1+1 (y) w 0 x T M 1 1 (x)[0.5 U N 1 +2 (y)+( N 1 +1) U N 1 (y)] w 0 y U N 1 +1 (y)[0.5 T M 1 (x)+( M 1 1) × T M 1 2 (x)]+ w 0 2 [0.5 T M 1 (x)+( M 1 1) T M 1 2 (x)][0.5 U N 1 +2 (y)+( N 1 +1) U N 1 (y)]} exp(i k 0 n o z){xy V M 1 1 (x) V N 1 +1 (y) w 0 x V M 1 1 (x)[0.5 V N 1 +2 (y)+( N 1 +1) V N 1 (y)] w 0 y V N 1 +1 (y)[0.5 V M 1 (x)+( M 1 1) V M 1 2 (x)]+ w 0 2 [0.5 V M 1 (x)+( M 1 1) V M 1 2 (x)] ×[0.5 V N 1 +2 (y)+( N 1 +1) V N 1 (y)]}},
E z (ρ,z)= (1) n i k 0 n o 2 2n+m+1 n!π z 2 exp(ie k 0 n o z) l=0 n s=0 [(m1)/2] (1) s ( n l )( m 2s+1 ){x T M 1 1 (x) U N 1 +1 (y) w 0 U N 1 +1 (y)[0.5 T M 1 (x)+( M 1 1) T M 1 2 (x)]}.

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